Because \(\mathcal {B}_{\text{mars}}^{d}\)  contains an uncountable number of
basis functions, we work with an
infinite-dimensional version of LASSO. Infinite-dimensional LASSO
formulations have been used in many papers including
[1]}, [2]}, [3]},
[4]}, [5]},
[6]}, [7]}, and
[8]}. The main idea behind
infinite-dimensional LASSO estimators is to
consider infinite linear combinations of basis functions that are
parametrized by finite signed measures and to measure complexity in
terms of the
variations of the involved signed measures. In the MARS context, by
infinite linear combinations of basis functions in \(\mathcal {B}_{\text{mars}}^{d}\) , we mean the
functions of the form
\(f_{a_{\mathbf {0}}, \lbrace \nu _{\alpha }\rbrace }(x_1, \dots , x_d) := a_{\mathbf {0}} + \sum _{\alpha \in \lbrace 0, 1\rbrace ^d \setminus \lbrace \mathbf {0}\rbrace }\int _{[0, 1)^{|\alpha |}} \prod _{j \in S(\alpha )} (x_j -t_j)_+ \, d\nu _{\alpha }(t^{(\alpha )}),\) 
